Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/138

116 $$\scriptstyle{\vdash:.p\supset q.\supset:p.r.\supset.q.r}$$

I.e. both sides of an implication may be multiplied by a common factor. This is called by Peano the "principle of the factor." It will be referred to as "Fact."

$$\scriptstyle{\vdash:.p\supset r.q\supset s.\supset:p.q.\supset.r.s}$$

I.e. if $$\scriptstyle{p}$$ implies $$\scriptstyle{q}$$ and $$\scriptstyle{r}$$ implies $\scriptstyle{s}$, then $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ jointly imply $$\scriptstyle{r}$$ and $$\scriptstyle{s}$$ jointly. The law of contradiction, "$\scriptstyle{\vdash.\sim(p.\sim p)}$," is proved in this number (*3·24); but in spite of its fame we have found few occasions for its use.

$$\scriptstyle{p.q.=.\sim(\sim p\or\sim q)\quad\text{Df}}$$

$$\scriptstyle{p\supset q\supset r.=.p\supset q.q\supset r\quad\text{Df}}$$

Given two asserted elementary propositional functions "$\scriptstyle{\vdash.\phi p}$" and "$\scriptstyle{\vdash.\psi p}$" whose arguments are elementary propositions, we have $\scriptstyle{\vdash.\phi p.\psi p}$.

Dem.

$$\scriptstyle{\vdash:p.q.\supset.\sim(\sim p\or\sim q)\quad[\text{Id}.(*3\cdot01)]}$$

$$\scriptstyle{\vdash:\sim(\sim p\or\sim q).\supset.p.q\quad[\text{Id}.(*3\cdot01)]}$$

$$\scriptstyle{\vdash:\sim p.\or.\sim q.\or.p.q\quad\left[*2\cdot11\frac{\sim p\or\sim q}{p}\right]}$$

$$\scriptstyle{\vdash:\sim(p.q).\supset.\sim p\or\sim q\quad[*3\cdot11.\text{Transp}]}$$

$$\scriptstyle{\vdash:\sim p\or\sim q.\supset.\sim(p.q)\quad[*3\cdot1.\text{Transp}]}$$

$$\scriptstyle{\vdash:.p.\supset:q.\supset.p.q\qquad[*3\cdot12]}$$

$$\scriptstyle{\vdash:.q.\supset:p.\supset.p.q\qquad[*3\cdot2.\text{Comm}]}$$

$$\scriptstyle{\vdash:p.q.\supset.q.p}$$

This is one form of the commutative law for logical multiplication, A more complete form is given in *4·3.

Dem.